where \(I\)  is the integral operator associated with the kernel \(\mathbf {k}\)  defined in (REF ), which is given by
\([I u](x) = \int _{\mathbb {R}^N}(u(x)-u(y))\mathbf {k}(x-y)\,dy\) . Consequently, we have \(u \in C_{loc}(\Omega )\)  by the regularity result [1]} of Kassmann and Mimica and a straightforward approximation argument. It thus remains to prove the estimate (REF ) for fixed \(\tau \in (0,\frac{1}{2})\) . Since \(\Omega \)  satisfies a uniform exterior sphere condition, there exists a radius \(0<R_0< \frac{1}{2}\)  such that for every point \(x_* \in \partial \Omega \)  there exists a ball \(B^{x_*}\)  of radius \(R_0\)  contained in \(\mathbb {R}^N \setminus \overline{\Omega }\)  and
tangent to \(\partial \Omega \)  at \(x_*\) . Let \(c(x_*)\)  denote the center of \(B^{x_*}\) . We now apply Lemma REF  with the value \(R: =\frac{R_0}{2}\) . This yields a function \(V \in L^1_0(\mathbb {R}^N)\)  and \(\delta >0\)  such that the properties (i)-(iv) of Lemma REF  are satisfied. By Lemma REF (iv), we may assume that \(\delta >0\)  is chosen sufficiently small so that
\(L_{\text{\tiny $\Delta \,$}}\!V(x) \ge \Vert f\Vert _{L^\infty } \quad \text{for $x \in A_{R,\delta }:= \lbrace x \in \mathbb {R}^N \:: R<|x| <R+\delta \rbrace $.}\) 
